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A353619 Side a of primitive integer-sided triangles (a, b, c) whose angle B = 3*C. 4
3, 35, 119, 112, 279, 20, 253, 539, 552, 91, 923, 533, 476, 1455, 224, 1504, 17, 799, 2159, 1513, 1476, 437, 1387, 3059, 2261, 1240, 3160, 4179, 2163, 748, 3212, 391, 1817, 5543, 3151, 4393, 5712, 1175, 2825, 7175, 5825, 2548, 5876, 189, 9099, 4077, 5859, 1736, 9352, 5768, 1189 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of the side c; hence, this sequence of sides a is not increasing.
In the case B = 3*C, the corresponding metric relation between sides is c*a^2= (b-c)^2 * (b+c).
Equivalently, length of side common to the two angles, one being the triple of the other, of a primitive integer-sided triangle.
For the corresponding primitive triples and miscellaneous properties and references, see A353618.
LINKS
The IMO Compendium, Problem 1, 46th Czech and Slovak Mathematical Olympiad 1997.
FORMULA
a(n) = A353618(n, 1).
EXAMPLE
According to inequalities between a, b, c, there exist 3 types of such triangles:
a < c < b with the smallest side a = 3 of the first triple (3, 10, 8).
c < a < b with the middle side a = 35 of the 2nd triple (35, 48, 27).
c < b < a with the largest side a = 539 of the 8th triple (539, 510, 216), the first of this type.
MAPLE
for b from 4 to 9000 do
for q from 2 to floor((b-1)^(1/3)) do
a := (b-q^3) * sqrt(1+b/q^3);
if a= floor(a) and q^3 < b and igcd(a, b, q)=1 and (b-q^3) < a and a < b+q^3 then print(a); end if;
end do;
end do;
CROSSREFS
Cf. A353618 (triples), this sequence (side a), A353620 (side b), A353621 (side c), A353622 (perimeter).
Cf. A343064 (similar, but with B = 2*C).
Sequence in context: A231645 A076376 A338787 * A133710 A130061 A368391
KEYWORD
nonn
AUTHOR
Bernard Schott, May 07 2022
STATUS
approved

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Last modified March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)